calculate cos 45 degrees
Check the final answer first, then review the worked steps.
Problem
calculate cos 45 degrees
Answer
\(\frac{1}{\sqrt{2}}\)
Step-by-step solution
- Identify the triangle: The value of $\cos 45^{\circ}$ is derived from an isosceles right triangle (a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle). In such a triangle, the sides are in the ratio $1:1:\sqrt{2}$.
- Apply the definition of cosine: The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.
- Calculate the value: For a $45^{\circ}$ angle, the adjacent side is $1$ and the hypotenuse is $\sqrt{2}$. Therefore, $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$.
- Compare with options: Looking at the provided choices, option F is $\frac{1}{\sqrt{2}}$. Note that this is also equivalent to $\frac{\sqrt{2}}{2}$ by rationalizing the denominator, but $\frac{1}{\sqrt{2}}$ is explicitly listed as option F.